#### Conference abstracts

Session A1 - Approximation Theory

July 10, 14:30 ~ 15:20

## Nonlinear $n$-term Approximation of Harmonic Functions from Shifts of the Newtonian Potential

### Pencho Petrushev

### University of South Carolina, USA - pencho@math.sc.edu

A basic building block in Classical Potetial Theory is the fundamental solution of the Laplace equation in ${\mathbb R}^d$ (Newtonian potential). Our main goal is to study the rates of nonlinear $n$-term approximation of harmonic functions on the unit ball $B^d$ from shifts of the Newtonian potential with poles outside $\overline{B^d}$ in the harmonic Hardy spaces. Optimal rates of approximation are obtained in terms of harmonic Besov spaces. The main vehicle in establishing these results is the construction of highly localized frames for Besov and Triebel-Lizorkin spaces on the sphere whose elements are linear combinations of a fixed number of shifts of the Newtonian potential.

Joint work with Kamen Ivanov (Bulgarian Academy of Sciences).